In this work, we propose and theoretically analyze a new scheme for generation of counterpropagating photon pairs in photonic crystal waveguides through the process spontaneous four-wave mixing. Using fundamental properties periodic Bloch modes standard waveguide, demonstrate how modal phase-matching can be reached between forward-propagating pump signal idler modes, degenerate non-degenerate p...

We introduce a new concept of the bounded rank (with respect to a positive constant) for unital C∗-algebras as a modification of the usual real rank. We present a series of conditions insuring that bounded and real ranks coincide. These observations are then used to prove that for a given n and K > 0 there exists a separable unital C∗-algebra Z n such that every other separable unital C∗-algebr...

The conjecture that every algebra norm || • || on C(X) is equivalent to the uniform norm arises naturally from a theorem of Kaplansky in 1949 that necessarily 11/11 > \f\x ( ƒ e C(X)): see [9, 10.1]. The seminal paper on the automatic continuity of homomorphisms from C(X) is the 1960 paper of Bade and Curtis [2] in which, for example, it is proved that there is a discontinuous homomorphism from...

In this paper we establish a direct connection between stable approximate unitary equivalence for ∗-homomorphisms and the topology of the KK-groups which avoids entirely C*-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this to...

We consider the semigroup Ext(A, B) of extensions of a separable C *-algebra A by a stable C *-algebra B modulo unitary equivalence and modulo asymptotically split extensions. This semigroup contains the group Ext −1/2 (A, B) of invertible elements (i.e. of semi-invertible extensions). We show that the functor Ext 1/2 (A, B) is homotopy invariant and that it coincides with the functor of homoto...

According to J. Feldman and C. Moore’s wellknown theorem on Cartan subalgebras, a variant of the group measure space construction gives an equivalence of categories between twisted countable standard measured equivalence relations and Cartan pairs, i.e., a von Neumann algebra (on a separable Hilbert space) together with a Cartan subalgebra. A. Kumjian gave a C∗-algebraic analogue of this theore...

We give the nuclear analogue of Dadarlat’s characterization of exact quasidiagonal C∗-algebras. Specifically, we prove the following: Theorem 0.1. Let A be a unital separable simple C∗-algebra. Then the following conditions are equivalent: i) A is nuclear and quasidiagonal. ii) A has the stabilization principle. iii) If π : A → M(A ⊗ K) is a unital, purely large ∗-homomorphism, then the image π...

In a previous paper, we showed that every strongly commuting pair of CP0-semigroups on a von Neumann algebra (acting on a separable Hilbert space) has an E0-dilation. In this paper we show that if one restricts attention to the von Neumann algebra B(H) then the unitality assumption can be dropped, that is, we prove that every pair of strongly commuting CP-semigroups on B(H) has an E-dilation. T...

The class of generic structures among those consisting of the measure algebra of a probability space equipped with an automorphism is axiomatizable by positive sentences interpreted using an approximate semantics. The separable generic structures of this kind are exactly the ones isomorphic to the measure algebra of a standard Lebesgue space equipped with an aperiodic measure-preserving automor...

Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations an...